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Properties

Label 855.2.da.b
Level $855$
Weight $2$
Character orbit 855.da
Analytic conductor $6.827$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(199,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.da (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 18 q^{4} + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 18 q^{4} + 6 q^{5} - 15 q^{10} + 12 q^{11} - 6 q^{14} - 42 q^{16} + 12 q^{19} - 42 q^{20} + 12 q^{25} - 12 q^{26} - 42 q^{31} - 36 q^{34} - 6 q^{35} + 66 q^{40} - 6 q^{41} + 6 q^{44} - 6 q^{46} + 12 q^{49} + 18 q^{50} - 36 q^{56} + 36 q^{59} + 48 q^{61} + 18 q^{65} - 123 q^{70} + 24 q^{71} - 84 q^{74} + 66 q^{76} + 48 q^{79} + 39 q^{80} - 84 q^{85} + 42 q^{86} + 12 q^{89} - 30 q^{91} - 72 q^{94} + 63 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1 −1.93197 + 0.340658i 0 1.73706 − 0.632239i 2.15043 − 0.612911i 0 0.586358 + 0.338534i 0.257316 − 0.148561i 0 −3.94576 + 1.91668i
199.2 −1.45670 + 0.256855i 0 0.176607 − 0.0642796i −0.658585 + 2.13688i 0 2.81448 + 1.62494i 2.32124 − 1.34017i 0 0.410490 − 3.28195i
199.3 −1.20303 + 0.212126i 0 −0.477108 + 0.173653i −2.06524 − 0.857189i 0 −3.28379 − 1.89590i 2.65299 − 1.53170i 0 2.66638 + 0.593130i
199.4 −0.240763 + 0.0424530i 0 −1.82322 + 0.663598i −0.0152217 + 2.23602i 0 1.68032 + 0.970136i 0.834240 − 0.481648i 0 −0.0912608 − 0.538996i
199.5 0.240763 − 0.0424530i 0 −1.82322 + 0.663598i 1.42562 − 1.72267i 0 −1.68032 − 0.970136i −0.834240 + 0.481648i 0 0.270105 − 0.475278i
199.6 1.20303 − 0.212126i 0 −0.477108 + 0.173653i −2.13306 − 0.670867i 0 3.28379 + 1.89590i −2.65299 + 1.53170i 0 −2.70844 − 0.354594i
199.7 1.45670 − 0.256855i 0 0.176607 − 0.0642796i 0.869056 − 2.06028i 0 −2.81448 − 1.62494i −2.32124 + 1.34017i 0 0.736759 − 3.22442i
199.8 1.93197 − 0.340658i 0 1.73706 − 0.632239i 1.25335 + 1.85179i 0 −0.586358 − 0.338534i −0.257316 + 0.148561i 0 3.05226 + 3.15062i
244.1 −1.72697 + 2.05812i 0 −0.906145 − 5.13900i 1.71358 − 1.43654i 0 −2.41018 + 1.39152i 7.48810 + 4.32326i 0 −0.00273678 + 6.00761i
244.2 −1.14717 + 1.36714i 0 −0.205786 − 1.16707i −1.67705 − 1.47903i 0 −3.67118 + 2.11955i −1.25953 − 0.727188i 0 3.94589 − 0.596068i
244.3 −1.04072 + 1.24028i 0 −0.107903 − 0.611947i 2.12183 + 0.705562i 0 1.93288 − 1.11595i −1.93303 − 1.11604i 0 −3.08333 + 1.89738i
244.4 −0.288852 + 0.344240i 0 0.312230 + 1.77075i 0.869891 + 2.05992i 0 −1.78470 + 1.03040i −1.47809 − 0.853374i 0 −0.960378 − 0.295561i
244.5 0.288852 − 0.344240i 0 0.312230 + 1.77075i −1.52197 + 1.63818i 0 1.78470 − 1.03040i 1.47809 + 0.853374i 0 0.124303 + 0.997111i
244.6 1.04072 − 1.24028i 0 −0.107903 − 0.611947i −2.23519 − 0.0626993i 0 −1.93288 + 1.11595i 1.93303 + 1.11604i 0 −2.40397 + 2.70701i
244.7 1.14717 − 1.36714i 0 −0.205786 − 1.16707i 2.08176 − 0.816246i 0 3.67118 − 2.11955i 1.25953 + 0.727188i 0 1.27221 − 3.78244i
244.8 1.72697 − 2.05812i 0 −0.906145 − 5.13900i −1.11892 − 1.93598i 0 2.41018 − 1.39152i −7.48810 − 4.32326i 0 −5.91682 − 1.04052i
289.1 −0.810919 + 2.22798i 0 −2.77422 − 2.32785i 0.323811 + 2.21250i 0 1.41749 + 0.818386i 3.32944 − 1.92225i 0 −5.19199 − 1.07271i
289.2 −0.795068 + 2.18443i 0 −2.60752 − 2.18797i 0.914172 − 2.04066i 0 0.124103 + 0.0716510i 2.82626 − 1.63174i 0 3.73085 + 3.61941i
289.3 −0.358233 + 0.984236i 0 0.691698 + 0.580404i −2.23415 − 0.0926215i 0 −2.37320 − 1.37016i −2.63320 + 1.52028i 0 0.891507 − 2.16575i
289.4 −0.0854197 + 0.234689i 0 1.48431 + 1.24548i 1.06599 − 1.96562i 0 3.42983 + 1.98021i −0.851670 + 0.491712i 0 0.370253 + 0.418078i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.da.b 48
3.b odd 2 1 95.2.p.a ✓ 48
5.b even 2 1 inner 855.2.da.b 48
15.d odd 2 1 95.2.p.a ✓ 48
15.e even 4 2 475.2.l.f 48
19.e even 9 1 inner 855.2.da.b 48
57.j even 18 1 1805.2.b.l 24
57.l odd 18 1 95.2.p.a ✓ 48
57.l odd 18 1 1805.2.b.k 24
95.p even 18 1 inner 855.2.da.b 48
285.bd odd 18 1 95.2.p.a ✓ 48
285.bd odd 18 1 1805.2.b.k 24
285.bf even 18 1 1805.2.b.l 24
285.bi even 36 2 475.2.l.f 48
285.bi even 36 2 9025.2.a.cu 24
285.bj odd 36 2 9025.2.a.ct 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.p.a ✓ 48 3.b odd 2 1
95.2.p.a ✓ 48 15.d odd 2 1
95.2.p.a ✓ 48 57.l odd 18 1
95.2.p.a ✓ 48 285.bd odd 18 1
475.2.l.f 48 15.e even 4 2
475.2.l.f 48 285.bi even 36 2
855.2.da.b 48 1.a even 1 1 trivial
855.2.da.b 48 5.b even 2 1 inner
855.2.da.b 48 19.e even 9 1 inner
855.2.da.b 48 95.p even 18 1 inner
1805.2.b.k 24 57.l odd 18 1
1805.2.b.k 24 285.bd odd 18 1
1805.2.b.l 24 57.j even 18 1
1805.2.b.l 24 285.bf even 18 1
9025.2.a.ct 24 285.bj odd 36 2
9025.2.a.cu 24 285.bi even 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 9 T_{2}^{46} + 78 T_{2}^{44} + 181 T_{2}^{42} - 255 T_{2}^{40} - 10179 T_{2}^{38} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display